A new metric of crime hotspots for Operational Policing
Monsuru Adepeju*1, Tao Cheng†1, John Shawe-Taylor‡2, Kate Bowers₸3
1
SpaceTimeLab for Big Data Analytics, Department of Civil, Environmental and Geomatic
Engineering, University College London
2
Department of Security and Crime Science, University College London
3
Department of Computer Science, University College London
November 07, 2014
Summary
This study examines the existing metrics used in evaluating the effectiveness of area-based crime
hotspots for operational policing. We identified some of the limitations of the metric (i.e. Area-toPerimeter (AP) ratio) used for measuring compactness of hotspots and then proposed a new improved
metric called “Clumpiness Index (CI)”. The case study of London Metropolitan police crime dataset
features the prediction of 3 different crime types using two different crime predictive methods. The
effectiveness of the hotspots was then measured using both AP ratio and CI. The comparison of the
results clearly shows that CI is a better metric for measuring the effectiveness of crime hotspots for
operational policing.
KEYWORDS: Effective hotspots, Area-to-perimeter (AP) ratio, Hit rate, Clumpiness Index (CI),
Operational Policing.
1. Introduction
As police resources are becoming increasingly limited due partly to budget constraints, there is a
growing interest in strategies that can enhance optimisation of their resources towards achieving the
desired crime prevention goals. An effective policing strategy is one that offers the police high crime
prevention potential with a small amount of police deployment (Weisburd, 2008). Over the last
decade, attempts to increase police effectiveness have resulted in operational policing being informed
by predictive analysis of crime. The predictive methods are used to identify locations of high future
crime risk. These locations are referred to as crime hotspots. The types of hotspots include pointbased, network-based and area-based hotspots. Several studies have suggested that police can be more
effective in intervening in crime by focussing on small geographical units with high crime rates
(hotpots) rather than actual people (offenders) committing the crimes (Telep & Weisburd, 2014). As a
result, most predictive methods of crime have been aimed at identifying area-based hotspots.
To estimate how effective the detected hotspots are for operational policing, two metrics have been
used, proposed by Bowers et al. (2004). These metrics are Hit rate (HR) and Area-to-Perimeter (AP)
ratio. The HR measures the proportion of future crime accurately captured by the purported hotspots
while AP ratio measures compactness (easiness of covering) the hotspots. Bowers et al. (2004)
recommended that the two measures should be used together for meaningful evaluation. This is
because hotspots with high HR may not necessarily be easily coverable based on their geometric
shape. Thus, hotspots with moderate HR and a high AP ratio are preferred to ensure effective
policing. However, certain limitations can be identified with the AP ratio which renders it less
appealing for evaluating hotspots for effective policing.
The AP ratio is used to measure the geometric complexities (compactness) of hotpots. The
*
monsuru.adepeju.11@ucl.ac.uk
tao.cheng@ucl.ac.uk
‡ j.shawe-taylor@ucl.ac.uk
₸
kate.bowers@ucl.ac.uk
†
assumption is that regular-shaped hotspots (e.g. squares) can be covered quicker and more easily than
irregularly-shaped hotspots, if we ignore the underlying network structure. The AP ratio has
limitations. They are:
a)
Holding the shape of a hotspot constant, the AP ratio varies with the spatial scale, (ranging from
zero to infinity). This makes it difficult to compare similar hotspots across different study areas.
b)
The level of disaggregation or dispersion of the hotspots (grid units) cannot be inferred from the
value of the AP ratio. Therefore, the AP ratio cannot give us an idea of randomly distributed
hotspots as a baseline for comparison.
c)
The AP ratio is relatively insensitive to differences in the structure of hotspots. Thus, although
hotspots may possess very different shapes, they may have identical area and perimeters.
Therefore, the goal of this paper is to propose a new metric for measuring effectiveness of crime
hotspots for operational policing. Specifically, we are proposing a new metric called Clumpiness
Index as an alternative to AP ratio given the limitations of AP ratio listed above.
2. Existing Metrics – Hit Rate and AP Ratio
2.1 Hit Rate: the proportion of new crimes captured by the defined hotspot. Evaluated at a certain
area coverage (e.g. 20% area coverage)
∑𝑚 (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑟𝑖𝑚𝑒𝑠)
𝐻𝑖𝑡 𝑅𝑎𝑡𝑒 = ( ∑𝑘=1
𝑛 (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑟𝑖𝑚𝑒𝑠) ) × 100
(1)
𝑖=1
Where i = number of ranked grids; k = number of percentile of ranked grids e.g. 20th; n = total
number of grids.
2.2 Area-to-perimeter (AP) ratio: a measure of how compact an identified cluster (hotspot) is. The
more compact a hotspot is, the easier and quicker it will be to cover operationally. Higher AP ratio
corresponds to better compactness (Figure 1).
AP ratio = 6/14 = 0.43
AP ratio = 6/10 = 0.6
Figure 1 Area-to-Perimeter (AP) Ratio. The hotspot on the right pane is more compact and therefore
has higher AP ratio and may be seen as more efficient in operational policing terms.
3. A new metric - Clumpiness Index (CI)
Provided a modest hit rate, the actual effectiveness of a predictive solution is measured in terms the
geometric complexity (compactness) and distribution of the hotspots across a geographical area. We
propose a new metric called “Clumpiness Index (CI)” which measures the compactness and
distribution of hotspots and is robust to scaling issues of AP ratio.
The Clumpiness Index (CI) was originally proposed by Turner (1989) as Contagion Index for
measuring the overall clumpiness of categorical patches on a landscape. CI is able to measure
effectively both patch type interspersion (i.e. the intermixing of units of different patch types) as well
as patch dispersion (i.e. the spatial distribution of a patch type) at the landscape level. CI is computed
by first summarising the adjacency of all cells in an adjacency matrix, which shows the frequency
with which different pairs of patch types (including adjacencies between the same patch type) appear
side-by-side on the map. CI is defined as follows:
𝐺𝑖 −𝑃𝑖
𝑔𝑖𝑖
𝐺𝑖 = (∑𝑚
𝑘=1 𝑔𝑖𝑘
1−𝑃𝑖
𝐺𝑖 −𝑃𝑖
) ; 𝐶𝐼 =
{
1−𝑃𝑖
𝑃𝑖 −𝐺𝑖
−𝑃𝑖
for 𝐺𝑖 ≥ 𝑃𝑖
(2)
for 𝐺𝑖 < 𝑃𝑖 ; 𝑃𝑖 ≥ 0.5
for 𝐺𝑖 < 𝑃𝑖 ; 𝑃𝑖 < 0.5
gii = number of like adjacencies (joins) between pixels of patch type (class) i
gik = number of adjacencies (joins) between pixels of patch types (classes) i and k
Pi = proportion of the landscape occupied by patch type (class) i.
The goal of CI is to determine the maximum value of gii for any Pi
The CI takes values between -1 (when the class is maximally disaggregated) to 1 (when the class is
maximally aggregated corresponding to a checkerboard arrangement).
4. Case Study
4.1 Camden Borough of London
Camden Borough is one of the 12 inner boroughs of London City with an estimated 224,962
inhabitants as of 2011. The population density is estimated as approximately 10,000 people per
square kilometre (2011 Census, Office of National Statistics). The Borough contains a mixture of
commercial and residential areas with the busiest parts being the Camden Market and Covent Garden
and Holborn. The borough recorded a crime rate of 145 crimes per 1,000 people in 2010/11, the
national average being 75 crimes per 1,000 people (Source: Metropolitan Police Service website,
2014).
4.2 Effectiveness of hotspot for operational policing
Three different crime types are used in this analysis. They are shoplifting, violence-against-persons
and burglary (in-dwellings) crimes. The data points are aggregated to a grid system of 250m by 250m
and have temporal resolution of 1 day. The time period predicted is between 28/09/2011 and
6/01/2012, predicting 2 days ahead using the crime risk surface produced on each day.
To check the predictions against the real dataset, we overlay future crime on the predictive surface
generated. For example, validating the prediction on day tn means overlaying crime data from day tn+1
to day tn+2 on the predictive surface generated on day tn. By so doing, the proportion of crimes that are
captured by the ranked top 20% of the grids squares (hotspots) is evaluated (assuming that police only
has resources to cover just 20% of the Camden).
Two hotspots predictive methods are used, namely (i) Prospective Hotspot (Bowers et al. 2004) and
(ii) Kernel Density Estimation (KDE) method. Figure 2 represents the average of percentage hit rate
over the prediction period. Also included in Figure 2 is baseline prediction which is generated by way
of picking grid squares with equal probability (random) until 20% coverage is attained. This is
represented with the green line. The general performance of these predictive methods in terms of hit
rates follows the spatial concentration of different crime types with highly spatially concentrated
crime type showing highest hit rates. For example, shoplifting crime is highly concentrated in a few
regions near commercial areas and therefore shows the highest hit rates whereas prediction of
burglary crimes is lowest as residential properties are dispersed across the entire borough.
Violence-Against-Persons
Burglary-in-dwellings
% Area Flagged
% Area Flagged
% Area Flagged
% Crime Predicted
Shoplifting
Figure 2 Average % hit rates over the prediction period
In measuring hotspot compactness, we adapted CI to crime hotspots by classifying grid squares
constituting the predictive surface into two types, namely (1) Hot Spot – the top 20% ranked grids
and (2) Cold Spot – the remaining 80%. Figure 3 shows examples of predictive surfaces generated by
Prospective Hotspot method and random grid selection to illustrate how AP ratio and CI vary with
different hotspot configurations. In each example, we consider two spatial scalings: the original data
(unchanged), and data scaled such that each grid square has unit length. The AP ratio is observed to
change at different scales of measurement of the same surface, making it difficult to compare.
However, CI remains the same at any given scale. This is because CI is based on the adjacencies as
well as the proportion of the hotspot grids across total surface. Therefore, CI is able to provide a sense
of dispersion from a complete disaggregation (CI = -1) of the hotspot units.
´´´
´
Predictive
Surface
1 1
Predictive
Surface
(Random
gridding)
(Random
gridding)
Predictive
Surface 1
(Random gridding)
At aAt
unit
grid
size:
Predictive
Surface
1grid:
aa unit
grid
size:
At
unit
AP ratio
= 0.19
(Random
gridding)
AP
ratio
= 0.19
At
a
unit
size:
AP
ratio
=grid
0.19
CI =CI
0=0
AP
ratio
= 0.19
CI =
-0.45
At
a 0unit
grid
size:
CI actual
=
At actual
gridgrid
measurement:
At
measurement:
AP
ratio
= 0.19
At ratio
actual
grid measurement:
AP ratio
= 68.02
AP
= 68.02
At =
actual
grid measurement:
CI
0
CI =CI
0=0
AP
ratio==68.02
68.02
AP
ratio
At
CI =actual
=0-0.45grid measurement:
CI
AP ratio = 68.02
CI = 0
Predictivesurface
surface
Predictive
22
(Prospective
Hotspot)
(Prospective
Hotspot)
Predictive surface
2
(Prospective Hotspot)
Ata a
unit
grid size:
At
unit
grid
PredictiveAt
surface
2 size:
aratio
unit
grid:
AP
ratio
= 0.37
AP
0.37
At ratio
a
unit==grid
size:
(Prospective
Hotspot)
AP
CI =
0.420.37
CI
0.42
AP= ratio
= 0.37
CICIAt
==0.32
a
unit grid size:
0.42
actual
grid
measurement:
AtAtactual
grid
measurement:
AP ratio = 0.37
AP
ratio
=95.31
95.31
AP
ratio
=grid
At
actual
AtCI
actual
gridmeasurement:
measurement:
=0.42
0.42
CI =
CI
0.42
AP
== 95.31
AP=ratio
ratio
95.31
At
actual
grid
measurement:
0.42
CICIAP
==0.32
ratio = 95.31
CI = 0.42
00
800
800
0
800
0
800
1,600
1,600
1,600
1,600
2,400 3,2003,200
2,400
Meters
Meters
2,400
3,200
Meters
2,400
3,200
Meters
Predictive
surface
3 3
Predictive
surface
(Prospective
Hotspot)
(Prospective
Hotspot)
Predictive surface
3
Predictive surface 3
(Prospective
Hotspot)
At aAt
unit
grid
size:
(Prospective
Hotspot)
a unit
grid
size:
AP ratio
= 0.66
AP ratio
= 0.66
At
aa
unit
grid
size:
At
unit
At
a
unit
grid
size:
CI =CI
0.88
= 0.88 grid:
AP
ratio
=
0.66
AP
AP ratio = 0.66
At actual
gridgrid
measurement:
CI actual
CI
= 0.88
At
measurement:
CI == 164.58
0.54
AP ratio
AP ratio
= 164.58
At
actual grid measurement:
measurement:
At
CI =CI
0.88
= 0.88
At
actual
grid measurement:
AP
ratio
=
AP
ratio
= 164.58
164.58
CI
=
0.88
AP
ratio
=
164.58
CI = 0.88
Legend
Legend
Legend
Legend
Legend
Legend
Legend
Hot
Spot
3 3Hot
Legend
Spot
3 3 Hot Spot
Hot
Spot
Cold
Spot
Cold
Spot
Cold
Spot
Cold Spot
AP
= Area-to-Perimeter
AP
= Area-to-Perimeter
CI= Clumpiness
= Clumpiness
Index
CI
Index
AP
AP= =Area-to-Perimeter
Area-to-Perimeter
CICI= =Clumpiness
Index
Clumpiness
Index
CI = 0.54
Figure 3 Evaluating hotspot compactness with AP ratio and CI
5. Discussion and Conclusion
This study examines the use of existing metrics for measuring the effectiveness of predictive hotspots
for operational policing. We highlighted some of the limitations of AP ratio, a metric that is
specifically designed to evaluate effectiveness of predictive hotspots. We then proposed a new metric
called Clumpiness Index CI which is able to eliminate the limitations of AP ratio. This study first
established that the two predictive methods used (i.e. Prospective Hotspot and KDE) are able to
predict crimes well above the baseline predictions (random), and their performances are observed to
vary according to the spatial concentration of different crime types. The CI was then used to provide
more interpretable assessment of hotspot compactness which is found to be very robust to change in
scales of spatial units of analysis. In addition, the CI calculation provided a baseline of comparison
i.e. either of maximally aggregated (CI = -1) or maximally disaggregated (CI = 1) hotspot
configuration, giving a sense how easy the hotspots can be covered by the police. As with AP ratio
however, the CI being a single-valued metric requires visualisation of the purported hotspot to make
perfect meaning out of it.
6. Acknowledgements
This research is part of the CPC (Crime, Policing and Citizenship) project supported by UK EPSRC
(EP/J004197/1), in collaboration with the London Metropolitan Police.
7. Biography
Monsuru Adepeju is currently 2nd year PhD student at the SpaceTimeLab for Big Data Analytics, at
University College London. His PhD focuses on predictive modelling of space-time hotspot of crime
and his research interests include validation of crime predictive models for predictive policing and
development of usable predictive tools for operational environment.
Tao Cheng is a Professor in GeoInformatics, and Director of SpaceTimeLab for Big Data Analytics
(http://www.ucl.ac.uk/spacetimelab), at University College London. Her research interests span
network complexity, Geocomputation, integrated spatio-temporal analytics and big data mining
(modelling, prediction, clustering, visualisation and simulation), with applications in transport, crime,
health, social media, and environmental monitoring.
John S Shawe-Taylor is a professor at University College London (UK) where he is co-Director of
the Centre for Computational Statistics and Machine Learning (CSML). His main research area is
Statistical Learning Theory, but his contributions range from Neural Networks, to Machine Learning,
to Graph Theory. He has coordinated a number of European wide projects investigating the theory
and practice of Machine Learning, including the NeuroCOLT projects.
Kate Bowers is a Professor in Crime Science at the UCL Department of Security and Crime Science.
Kate has worked in the field of crime science for almost 20 years, with research interests focusing on
the use of quantitative methods in crime analysis and crime prevention.
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Turner M G (1989). Landscape ecology: the effect of pattern on process. Annual Review of Ecology
and Systems, 20: 171–197
Weisburd D (2008). Place-based policing, Ideas in American policing. Police Foundation,
Washington, DC. http://www.policefoundation.org/content/place-based-policing. Accessed on
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